import numpy as np
import plotly.express as px
import plotly.io as pio
import plotly.offline as py
pio.renderers.default = "notebook"
v = np.array([[1,0,3], [-1,0,3]])
v = v/np.linalg.norm(v, 2, axis=1)[:, np.newaxis]
n = 3000
np.random.seed(1)
x = np.random.normal(size=(n,3))
x = x / np.linalg.norm(x, 2, axis=1)[:, np.newaxis]
x = x[np.dot(x,v[0])*np.dot(x,v[1]) <0, :]
print(f'v1 = {np.round(v[0],3)}, \nv2 = {np.round(v[1],3)}')
print(f'arccos(v1,v2)/pi = {np.round(np.arccos(v[0] @ v[1].T)/np.pi,3)}')
print(f'simulated result = {np.round(len(x)/n,3)}')
fig = px.scatter_3d(x=x[:,0], y=x[:,1], z=x[:,2], size=np.ones(len(x)), range_z=[-1,1])
fig.add_scatter3d(x=[0, v[0,0]], y=[0, v[0,1]], z=[0, v[0,2]], name='v1')
fig.add_scatter3d(x=[0, v[1,0]], y=[0, v[1,1]], z=[0, v[1,2]], name='v2')
fig.show()